I heard this song "Father & Son" by Cat Stevens. I really liked it. I liked this fact that Cat used a deeper register for the father while a higher one for the son. What is the genre of this song or what type of song is this called, with a mix of deeper and higher registers like this? How can I find other songs like this?
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This isn't really a genre, because Stevens uses the two registers for a very specific reason based on the lyrics --he's playing two different characters, one older and one younger. The only other songs I can think of that do characters by register are the middle section of Queen's "Bohemian Rhapsody,", and the goofy call-and-response section of Justin Timberlake's Senorita which are both very different songs. Billy Joel plays two different characters in "Still Rock and Roll to Me," but they are in the same register.
There are duets, usually between male and female singers, that have a similar effect, but those don't necessarily cluster in any one genre.
I hope this is the type of thing you're after https://youtu.be/R9IWAxSpgKA